Question

the coordinates of the endpoints of \\(\overline{lm}\\) are \\(l(-7, -7)\\) and \\(m(5, 9)\\). point \\(n\\) is on \\(\overline{lm}\\) and divides it such that \\(ln:mn\\) is \\(1:3\\). what are the coordinates of \\(n\\)? write your answers as integers or decimals. \\((\square, \square)\\)

Explanation:

Step1: Recall the section formula

The section formula for a point \( N(x,y) \) dividing the line segment joining \( L(x_1,y_1) \) and \( M(x_2,y_2) \) in the ratio \( m:n \) is given by:
\( x=\frac{mx_2 + nx_1}{m + n} \) and \( y=\frac{my_2 + ny_1}{m + n} \)
Here, \( L(-7,-7) \), \( M(5,9) \), \( m = 1 \), \( n = 3 \)

Step2: Calculate the x - coordinate of N

Substitute \( x_1=-7 \), \( x_2 = 5 \), \( m = 1 \), \( n = 3 \) into the formula for \( x \):
\( x=\frac{1\times5+3\times(-7)}{1 + 3}=\frac{5-21}{4}=\frac{-16}{4}=-4 \)

Step3: Calculate the y - coordinate of N

Substitute \( y_1=-7 \), \( y_2 = 9 \), \( m = 1 \), \( n = 3 \) into the formula for \( y \):
\( y=\frac{1\times9+3\times(-7)}{1 + 3}=\frac{9-21}{4}=\frac{-12}{4}=-3 \)

Answer:

\((-4, -3)\)